Question $12(7$ points$)$

A company has found that the rate at which a person new to an assembly line produces items is given by:

$\frac{dy}{dt}=12e^{-0\mathrm{.}25y}$

where $y$ represents the number of items and $t$ represents the number of days the person has worked on the line.

Solve the differential equation if $x(0)=0.$

How many items should the worker be expected to produce on the ${10}^{th}$ day?

When should the new worker be expected to produce $20$ items per day?

According to this model, will the new worker ever be able to produce $50$ items per day? Justify your answer.